On Efficient Noncommutative Polynomial Factorization via Higman Linearization

• Published in: Computational complexity Vol. 34; no. 2; p. 10
• Main Authors: Arvind, V., Joglekar, Pushkar S.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2025; Springer Nature B.V
• Subjects: Algorithm Analysis and Problem Complexity; Algorithms; Computation; Computational Mathematics and Numerical Analysis; Computer Science; Factorization; Fields (mathematics); Linearization; Mathematical analysis; Polynomials; Rings (mathematics); identity testing; Higman linearization; 68 Computer Science; invariant subspaces; factorization; Noncommutative polynomials; linear matrices
• ISSN: 1016-3328; 1420-8954
• DOI: 10.1007/s00037-025-00272-9

In this paper we study the problem of efficiently factorizing polynomials in the free noncommutative ring F < x 1 , x 2 , … , x n > of polynomials in noncommuting variables x 1 , x 2 , … , x n over the field F . We obtain the following result: Given a noncommutative algebraic branching program of size s computing a noncommutative polynomial f ∈ F < x 1 , x 2 , … , x n > as input, where F = F q is a finite field, we give a randomized algorithm that runs in time polynomial in s , n and log 2 q that computes a factorization of f as a product f = f 1 f 2 ⋯ f r , where each f i is an irreducible polynomial that is output as a noncommutative algebraic branching program. The algorithm works by first transforming f into a linear matrix L using Higman linearization of polynomials. We then factorize the linear matrix L and recover the factorization of f . We use basic elements from Cohn's theory of free ideals rings combined with Ronyai's randomized polynomial-time algorithm for computing a nontrivial common invariant subspace of a collection of matrices over finite fields.